13304 PRODUCTION ENGINEERING I PETROLEUM ENGINEERING FACULTY OF EXPLORATION AND PRODUCTION TECHNOLOGY

13304

PRODUCTION ENGINEERING I

PETROLEUM ENGINEERING

FACULTY OF EXPLORATION AND PRODUCTION TECHNOLOGY

GAS WELL

• Darcy’s Law for Oil flow is also used in equation of gas flow

• The solution of partial differential equation from combination of The Continuity and Darcy’s Law for radial flow

• The unit of variables:

• P_r : reservoir pressure, psia

• P_wf : flowing bottom hole pressure, psia

• k = permeability, md

• h = formation thickness, ft

• T_R : reservoir temperature, ^oR

GAS WELL

Temperature

P_r

P_wf

gas reservoir

Pressure Specific assumptions:

➢ the compressibility and the viscosity of the fluid can’t be considered as constant

➢ the flow rate is high → turbulence

→ more pressure losses

➢ the liquid fraction is neglected

Basic Equation Gas Flow in Porous Medium

• Derivation of Darcy’s law for gas in radial flow.

• Similar single model as mentioned in Darcy’s Law for Oil flow is also used in this derivation

• The first equation on the left is Darcy’s Law for gas flow

• In this equation the main assumption is the gas flow is laminar

• The unit of variables:

• P_r : reservoir pressure, psia

• P_wf : flowing bottom hole pressure, psia

• k = permeability, md

• h = formation thickness, ft

• T_R : reservoir temperature, ^oR

• Z : gas compressibility

• r_e : draiange radius, ft

• r_w : wellbore radius, ft

• q_sc : gas production rate, MMSCF/d

• C : performance constant of well

( )



 



 −



 



 

= ⁻ −

75 . r 0

ln r Z T

P P

kh 10

x 703 . q 0

w e R

g

2 wf 2

r 6

sc



 



 −



 



 

= ⁻

75 . r 0

ln r Z T

kh 10 x 703 . C 0

w e R

g

6

(

wf²

)

2 r

sc

C P P

q = −

Gas Flow Equation – Turbulent Flow

• Based on empirical observations,

Rawlins and Schellhardt modified the equation, by adding the exponent “n”

that shows deviation from the ideal flow behavior.

• Refer to the equation for non-ideal condition, the relationship of q_sc vs (P_r²-P_wf²) would develop straight line in a log-log plot.

• The slope of the plot would be equal to 1 in laminar flow, and less than 1 in turbulent flow.

• The minimum value of n is 0.5

(

_r² _wf²

)

sc

C P P

q = −

(

_r² _wf²

)

ⁿ

sc

C P P

q = −

Ideal – Laminar Flow

Non - Ideal – Turbulent Flow The values of C and n are obtained using test data

Back Pressure Test

• The first method of test to determine

productivity of gas wells is Back Pressure Test

• The diagram on the left show how the test is conducted

• The first step, the well is shut in until the pressure in the reservoir reach reservoir pressure

• Then the well is produced at a certain rate (by applying a certain choke size), and the test is run until a constant production rate is obtained.

At this flow period the bottom hole flowing pressure is measured

• The above procedure are repeated 4 times, and the result could be plotted in log-log paper.

Isochronal Test

• The procedure is quite similar to Back Pressure Test, unless in isochronal test, the well is shut in before changing the flow rate.

• The production period and shut in

period are conducted at certain period of time, and this step is repeated 4 times, at different flow rate.

• This test represent transient conditions

• At the end of test, prolong production test is conducted to obtain stabilized pressure. This test show stabilized deliverability line

Modified Isochronal Test

• Similar procedure to Isochronal

test is conducted, unless the period time of production and shut in are conducted in the same time. The test are concluded by conducting extended flow rate.

• The data interpretation is similar to isochronal test, unless the value of Pr is taken from the data at every shut in condition.

• In a case of a gas well, the IPR is a curve. Mainly two models can be used to represent the behaviour of the gas flowing in the reservoir : the 2 back pressure equations.

• The parameters of these models can be determined with help from isochronal well test results. The most adequate model is the one which is the closest to the measurements.

• The default model is the second back pressure equation. In this equation, n is all the more close to 0.5 that the flow is turbulent.

IPR curve (gas well)

MODELS

2 parameters to characterize the well behaviour:

(a ; b) or (C ; n) determined from well tests

In the case of stabilized high flow rates, 2 main types of gas well behaviours:

( )

0

2 2

− =

− +

g wf r

q P b P

aq_g

First back pressure Equation Second back pressure Equation

(

r wf

)

ⁿ

g C P P

q = ² − ²

KATZ’S TEST

t_1f t_2i t_3i t_3f Time

P_wf

t_2f

t 

t

t

P_rm

t_4i t_4f

t

q

q₁ q₂ q₃ q₄ q₅

P_wf1

P_wf2

P_wf3

P_wf4 P_wf5

t_1i P_{wf initial} = P_r

t_bu t_bu t_bu

In this test, P_wfi and q are unstable values

stabilized pressure

IDENTIFICATION OF BOTH BACK PRESSURE EQUATIONS FROM WELL TESTS

( )

0

2 2

− =

−

+ q

P b P

aq ^{r wf}

can be written as linear functions

First Back Pressure Equation

( )

q P b P

aq ^{r wf}

2 2 −

= +

(

^{2 2}

)

log log

log q_g = C + n P_r − P_wf

(

_{r wf}

)

ⁿ

g C P P

q = ² − ²

Second Back Pressure Equation

CASE OF STABILIZED WELL TEST

(

^{2 2}

)

log log

logq_g = C + n P_r − P_wf

(

^{2 2}

)

log P_rm − P_wf

log q_g

logC

n = slope of the linear regression

logC= intersection between the linear regression and the logq axis

log-log plot

example of the identification of the second back pressure equation

n

CASE OF NON STABILIZED WELL TESTS

(

^{2 2}

)

log P_rm − P_wf

log q

logC

point obtained

with (P_wf5,q₅) = stabilized point points obtained

during drawdown periods (P_wfi,q_i) , i = 1..4

(

r wf

)

ⁿ

g C P P

q = ² − ²

example of the identification of the second back pressure equation

METHOD TO IDENTIFY MODEL PARAMETERS WITH WELL TESTS MEASUREMENTS

( )22logwfrm PP−( )22logwfrm PP− ( )22logwfrm PP−( )22logwfrm PP−

With well tests, we measure 4 or 5 times q and P_wf

Back Pressure 1

( )

q P P_r² − _wf²

We calculate

model 1

We plot q versus

( )

q P P_r² − _wf²

linear regression (+ use of stabilized (q,P_wf))

a and b determination

Back Pressure 2

We calculate log q and

model 2

We plot log q versus

linear regression (+ use of stabilized (q,P_wf))

n and logC determination

(

^{2 2}

)

log P_rm −P_wf

(

^{2 2}

)

log P_rm − P_wf

ABSOLUTE OPEN FLOW POTENTIAL

(

PrShutIn

)

ⁿ

C AOFP = ²

(

r wf

)

ⁿ

g

C P P

q =

²

−

²

AOFP represents the case of production where P_wf = 0.

In this case, P₁ is maximum, because :

wf rShutIn P P

P = −

 ₁

0

Then, the production rate is maximum

(by considering only the reservoir point of view).

example : The 2nd back pressure equation : can be written :

INFLOW – EXERCISE7 : MODELING OF A GAS WELL BEHAVIOUR

Flow rate (Mcfd) Bottom hole pressure (psia)

Shut in 3120

7800 2870

10590 2750

13960 2588

17615 2389

A dry gas well was tested at various flow rates with back pressure tests :

questions:

•By using the back pressure equations, build both models and give the AOFP of the well. Choose the adequate model.

•The well is flowed at 25% of the AOFP. In this case, what is the bottom hole pressure?

•The reservoir pressure declines to 2980 psia, what is the new AOFP ?

Back pressure 1: (P_r² – P_wf²) / q = aq + b

plot (P_r² – P_wf²)/q versus q and directly determine a and b

Back pressure 2: q = C (P_r² – P_wf²)ⁿ => log q = log C + n log (P_r² – P_wf²) plot log q vs. log (P_r² – P_wf²) and directly determine n and logC, hence C

Plot Test Data Pr (psi) = 3120

X mod1 q (Mcfd) 7800 10590 13960 17615

Pwf (psi) 2870 2750 2588 2389

Pr² - Pwf² 1497500 2171900 3036656 4027079

Y mod1 (Pr² - Pwf²)/q 192 205 218 229

X mod2 log(Pr² - Pwf²) 6.18 6.34 6.48 6.60

Y mod2 log(q) 3.89 4.02 4.14 4.25

mod1 = Back pressure 1 model plot mod2 = Back pressure 2 model plot

INFLOW – EXERCISE : MODELING OF A

GAS WELL BEHAVIOUR

INFLOW – EXERCISE : MODELING OF A GAS WELL BEHAVIOUR

21 21

Back pressure 1 model: does not yield a perfect fit

Plot determines directly (P_r² - P_wf²)/q = 0.0037q + 164.5 q @ Pwf = 0 => AOFP = 33.7 MMscfd

Back pressure 2 model achieves a better fit with test data in this case y = 0.8236x –1.1937 => log q = 0.8236 log (P_r² – P_wf²) –1.1937

n = 0.8236 and logC =-1.1937 => C = 0.064 Back pressure 1 model

y = 0,0037x + 164,5

190 200 210 220 230

5000 10000 15000 20000

q

(Pr2 - Pwf2)/q

Data

Linear (Data)

Back pressure 2 model

y = 0,8236x - 1,1937

3,8 3,9 4,0 4,1 4,2 4,3

6,1 6,2 6,3 6,4 6,5 6,6 6,7

Log (Pr2 – Pwf2)

Log q

Data

Linear (Data)

(

^{31202 2}

)

^0.8236

064 .

0 P_wf

q = −

IPR relationship:

INFLOW – EXERCISE : MODELING OF A GAS WELL BEHAVIOUR

• Absolute Open Flow Potential

( )

^{MMscf d}

AOFP= 0.064 9734400^0.8236 = 36.4 /

• P_wf for the well flowing at 25% AOFP

• AOFP after depletion

d MMscf q = 0.25*36.4 = 9.1 /

C psia P q

P_{wf r} ⁿ 2815

064 . 0

3120 9114 ^0.8236

1 2

1

2  =



 



−

 =



 



−

=

( )

^{MMscf d}

AOFP = 0.064 2980^{2 0.8236} =33.8 /

Isochronal Test Plot Exercise

Flow Test hours

Pwf psia

Q MMscf/d

Pr psi

Shut in 2200 0.00 2200

6 1892 2.80 2200

6 1782 3.40 2200

6 1647 4.80 2200

6 1511 5.40 2200

C = 4.29 x 10^-6 n = 0.94

AOF = 8.25 MMscf/d

Modified Isochronal Test Exercise

Time of Test (hrs)

Pwf, psia Flow Rate MMscf/d

Remarks

14 2000 0.00 Shut in

10 1842 4.00 Flow #1

10 1982 0.00 Shut in

10 1712 6.00 Flow #2

10 1960 0.00 Shut in

10 1511 8.00 Flow #3

10 1913 0.00 Shut in

10 1306 10.00 Flow #4

26 1072 10.00 Extended

Flow

68 2000 0 Final Shut

In

n = 0.76

C = 0.000124

AOF = 12.91 MMscf/d

IPR Gas Well

• In the case of gas wells, the velocity of the flow generates turbulences, which are represented in the models by a specific skin.

• Consequently, the relationship between the production rate and the drawdown isn't linear.

• We dispose on different models, and more particularly the two back pressure equations.

These models are generic, and can be adapted to each case of well by estimating their 2 parameters (a and b, or C and n) with help from well test analysis.

• The model used by default is the second back pressure equation, which is more often the most representative of the actual behaviour of the gas well. In this model, n is the factor of turbulence : when it is close to 0.5, the flow is very turbulent. When it is close to 1, the

turbulences are very low.

25

INFLOW PERFORMANCE RELATIONSHIP

P_wf

(psi)

P_r

q

(Mscf/day)

0

not linear – mainly due to turbulence

case of gas wells

AOFP

Reservoir Deliverability

• Reservoir pressure

• Pay zone thickness and permeability

• Reservoir boundary type and distance

• Wellbore radius

• Reservoir fluid properties

• Near-wellbore condition

• Reservoir relative permeability

28

THE RESERVOIR WELLBORE INTERFACE

Data:

reservoir thickness : 25 ft

reservoir permeability: 120 mD viscosity: 2.5 cP

FVF: 1.25 bbl/STB

well radius: 0.25 ft skin: 0

production rate: 600 STB/d

dP P f r S

r

h k q C

r

wf

P

P P w

e

=







 − +

 



=  ( )

4 ' ln 3

. 









 − +

 



= 

− '

4

ln 3 S

r r Ckh

P qB P

w e o

o wf

r



case of oil well one phase flow

Question:

Calculate the pressure profile and list the pressure drop across the following 1 ft intervals: [rw;1.25] [4;5] [19;20]

[99;100] [744;745]. Conclusion ?

INFLOW EXERCISE1 : NEAR WELLBORE PRESSURE PROFILE

( )( )( )

( )( )( ) ^_^{ }_^

+

= ln 0.25

25 120 00708 .

0

600 25

. 1 5 .

1800 2 r

P



 

 + 

=

w o

o

wf r

r Ckh

P qB

P  ln



 

 + 

=1800 88.28ln 0.r25 P

r (ft) p (psi) radius interval

pressure drop (psi)

0.25 1800

1.25 1942

4 2045

5 2064

19 2182

20 2186

99 2328

100 2329

744 2506.1

745 2506.2

744ft - 745ft

142

19

4

1

0.1 0.25ft - 1.25ft

4ft - 5ft

19ft - 20ft

99ft - 100ft

pressure profile (psi)

0 500 1000 1500 2000 2500 3000

0 100 200 300 400 500 600 700 800

radius (ft)

pressure (psi)

p (psi)

logarithmic shape

Conclusion

The near wellbore area plays a major role on the well productivity.

SKIN EFFECT ON THE PRESSURE DROP

near wellbore zone

near wellbore zone

P P_R

radius

P_wf

Estimated pressure profile without disturbance

P_skin > 0

Actual pressure profile in the case of a positive skin factor

P_skin < 0 Actual pressure profile in the case of

a negative skin factor

( )

_{wf nodisturb}

( )

_{wf Actual}

skin P P

P = −



well

well

reservoir

Well Performance Analysis

Well Performance Analysis

P_r, P_s, Q_p

IPR

VLP

Well

deliverability

Natural Flow well ?

Yes, but … no

Artificial lift

Q_p

yes

Artificial Lift

(start/restart, optimize)